Integrate 1/x from 1 to e^2: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int_{1}^{{e}^{2}} \frac{1}{x}\,dx$
2. 2. Steps, part 1
   1. $$\int_{1}^{{e}^{2}} \frac{1}{x}\,dx$$
      Start with the definite integral
   2. $$\int {x}^{-1}\,dx = \ln\left(\left| x\right|\right)$$
      Reciprocal rule: ∫ 1/x dx = ln|x|
   3. $$\int 1{x}^{-1}\,dx = \ln\left(\left| x\right|\right)$$
      Constant multiple rule: ∫ k·f dx = k∫ f dx
   4. $$F(x) = \ln\left(\left| x\right|\right)$$
      Antiderivative
3. 3. Steps, part 2
   1. $$F({e}^{2}) - F(1) = 2 - 0 = 2$$
      Apply the fundamental theorem of calculus
4. 4. The answer
   $$\int_{1}^{{e}^{2}} \frac{1}{x}\,dx = 2$$

   ∫ from 1 to e^2 = 2

5. 5. Check
   • ✓Differentiated the antiderivative and compared it with the integrand ✓
   • ✓Adaptive Simpson's rule gives 2 ✓

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